The electrostatic potential energy between proton and electron separated by a distance 1 Å is

Options

A. 13.6 eV
B. 27.2 eV
C. `-14.4 eV`
D. 1.44 eV

Answer

To find the electrostatic potential energy between a proton and an electron separated by a distance of 1 Å (1 angstrom), we can use the formula for the potential energy (U) between two point charges:

\[ U = \frac{k \cdot Q_1 \cdot Q_2}{r} \]

where:
- \( U \) is the electrostatic potential energy,
- \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges,
- \( r \) is the separation distance.

### Step-by-Step Solution:

1. **Identify the charges:**
- The charge of the electron (\( Q_e \)) is \( -1.6 \times 10^{-19} \, \text{C} \).
- The charge of the proton (\( Q_p \)) is \( +1.6 \times 10^{-19} \, \text{C} \).

2. **Convert the distance:**
- The distance \( r \) is given as 1 Å, which is \( 1 \, \text{Å} = 10^{-10} \, \text{m} \).

3. **Substitute the values into the formula:**
- Using the values in the formula:
\[ U = \frac{(9 \times 10^9) \cdot (-1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19})}{10^{-10}} \]

4. **Calculate the numerator:**
- Calculate \( 9 \times 10^9 \times (-1.6 \times 10^{-19}) \times (1.6 \times 10^{-19}) \):
\[ = 9 \times 10^9 \times (-2.56 \times 10^{-38}) = -23.04 \times 10^{-29} \]

5. **Divide by the distance:**
- Now divide by \( 10^{-10} \):
\[ U = \frac{-23.04 \times 10^{-29}}{10^{-10}} = -23.04 \times 10^{-19} \, \text{J} \]

6. **Convert to electron volts:**
- To convert joules to electron volts, we use the conversion \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ U = \frac{-23.04 \times 10^{-19}}{1.6 \times 10^{-19}} = -14.4 \, \text{eV} \]

### Final Answer:
The electrostatic potential energy between a proton and an electron separated by a distance of 1 Å is \( -14.4 \, \text{eV} \).